It has not unfrequently of late been made a subject of reproach to mathematicians who have occupied themselves with the theory of the refracting telescope, that the practical benefit derived from their speculations has been by no means commensurate to the expenditure of analytical skill and labour they have called for, and that from all the abstruse researches of Clairaut, Euler, D'Alembert, and other celebrated geometers, nothing hitherto has resulted beyond a mass of complicated formulæ, which, though confessedly exact in theory, have never yet been made the basis of construction for a single good instrument, and remain therefore totally inapplicable, or at least unapplied, in practice. The simplest considerations, indeed, suffice for the correction of that part of the aberration which arises from the different refrangibility of the differently coloured rays; and accordingly, this part of the mathematical theory of refracting telescopes was soon brought to perfection, and has received no important accession since the original invention of the achromatic object-glass. Indeed the theoretical considerations advanced on this part of the subject by Euler and D'Alembert have even had a tendency to retard its advancement, by appearing to establish relations among the relative refractive powers of media on rays of different colours which later experimental researches have exploded.